ABSTRACT (original article): The behavior of the kink in the sine-Gordon (sG) model in the presence of periodic inhomogeneity is studied. An ansatz is proposed that allows for the construction of a reliable effective model with two degrees of freedom. Effective models featuring very good agreement with the original field-theoretic partial differential equation are constructed, including in the nonperturbative region and for relativistic velocities. The numerical solutions of the sG model describing the evolution of the kink in the presence of a barrier as well as in the case of a periodic heterogeneity under the potential additional influence of a switched bias current and/or dissipation were obtained. The results of the field equation and the effective models were favorably compared. The effect of the choice of initial conditions in the field model on the agreement of the results with the effective model is also discussed. CITATION (original article): Tomasz Dobrowolski, Jacek Gatlik, Panayotis G. Kevrekidis (2025), Kink movement on a periodic background, Phys. Rev. E 111, 024203. https://doi.org/10.1103/PhysRevE.111.024203
The first part contains the calculations for the full sine-Gordon model modified by the space-dependent function F(x).We consider the function of the following form​
2
∂
t
ϕ-
∂
x
(F(x)
∂
x
ϕ)+sinϕ0
,where the function F(x) has the following form​
F(x)1+ϵg(x)
,
g(x)sin
π
12
x
.
ϵ=0.1;​​xp=-6;(*Initialpositionofkink*)​​tmax=400;​​α=0.02;(*Disipation*)​​v0=0;(*Initialvelocity*)​​lx=-50;(*Systemboundaryontheleft*)​​rx=50;(*Systemboundaryontheright*)
In[]:=
Γ[t_]:=Piecewise[{{0.015,50<t<200}},0](*Time-dependentbiascurrent*)
In[]:=
fF[x_]:=1+ϵ*Sin
π
12
*x
In[]:=
n=1;​​Monitorfullsol=NDSolve​​
∂
t,t
ϕ[x,t]+α
∂
t
ϕ[x,t]-
∂
x
(fF[x]*(
∂
x
ϕ[x,t]))+Sin[ϕ[x,t]]-Γ[t],​​ϕ[x,0]4ArcTanExp
1
fF[xp]
*
x-xp
1-
2
v0
,​​
(0,1)
ϕ
[x,0]-
2*v0
fF[xp]
*
1
1-v0
*Sech
1
fF[xp]
*
x-xp
1-
2
v0
,​​ϕ[lx,t]==0,​​ϕ[rx,t]==2*π​​,ϕ,{x,lx,rx},{t,0,tmax},MaxStepsInfinity,MaxStepSize0.1,StepMonitor:>(p=t;++n),{n,p};
The next part contains the functions that calculate the positions of the centers of kinks at each successive time step and fit the solution to the function
ϕ(x)4arctanexp
1
F(
x
0
)
γ(x-
x
0
)
. The center of the kink is defined as the point where
ϕ(t,x)π
.
In[]:=
xlimit=xp-1;​​outx0={};​​outγ={};​​dx=0.001;​​tempx=0;​​MonitorDo​​Do​​IfEvaluate[ϕ[xc,t]/.fullsol][[1]]>Pi,​​​​tempx=FindRoot[Evaluate[ϕ[x,t]/.fullsol][[1]]==π,{x,(2xc-dx)/2}][[1,2]],​​AppendTo[outx0,{t,tempx}],​​fit=NonlinearModelFitTable[{x,ϕ[x,t]/.fullsol[[1]]},{x,lx,rx,dx}],4ArcTanExp
1
fF[tempx]
*γ*(x-tempx),γ,x,​​AppendTo[outγ,{t,fit[[1,2,1,1,2]]}],​​Break[]​​​​,{xc,xlimit,rx,dx};​​xlimit=tempx-1;​​,{t,0,tmax,0.5},t
The second part contains a solution to the proposed effective model with two degrees of freedom, in which the equations of motion are defined as​
M
̈
x
0
-κ
̈
γ
+
1
2

∂
x
0
M
2

x
0
-
1
2

∂
x
0
m
2

γ
-(
∂
γ
κ)
2

γ
+(
∂
γ
M)

γ

x
0
+
∂
x
0
V0,
​​
m
̈
γ
-κ
̈
x
0
+
1
2
(
∂
γ
m)
2

γ
-
1
2
(
∂
γ
M)
2

x
0
-
∂
x
0
κ
2

x
0
+
∂
x
0
m

x
0

γ
+
∂
γ
V0.
​Where the functions ​
M
2
π
6γ
F(
x
0
)
2
∂
x
0
F(
x
0
)
F(
x
0
)
+
8γ
F(
x
0
)
,
​​
m
2
2
π
3
3
γ
F(
x
0
)
,
​​
κ
2
π
3
2
γ
1
F(
x
0
)
∂
x
0
F(
x
0
),
​​
V
4
F(
x
0
)
γ
+
2
2
γ
F(
x
0
)
+∞
∫
-∞
d
2
x
ξF(x).
In[]:=
ζ[x_,x0_,γ_]:=
1
fF[x0]
*γ*(x-x0);​​fM[x0_?NumericQ,γ_?NumericQ]:=
2
π
6γ
fF[x0]
2
(D[fF[xx],xx]/.xxx0)
fF[x0]
+
8γ
fF[x0]
;​​fm[x0_?NumericQ,γ_?NumericQ]:=
2
2
π
3
3
γ
fF[x0]
;​​fκ[x0_?NumericQ,γ_?NumericQ]:=
2
π
3
2
γ
*
1
fF[x0]
*(D[fF[xx],xx]/.xxx0);​​fV[x0_?NumericQ,γ_?NumericQ]:=
4
fF[x0]
γ
+
2
2
γ
fF[x0]
*NIntegrate
2
Sech[ζ[x,x0,γ]]
*fF[x],{x,lx,rx};
In order to speed up the calculations, we calculate the values of the function V before running NDSolve.
fcV=ListInterpolation[ParallelTable[Re[fV[x0,γ]],{x0,-20,25,0.1},{γ,0.5,1.5,0.025}],{{-20,25},{0.5,1.5}}];
In[]:=
Clear[t];​​n=1;​​Monitormodel=NDSolve​​
∂
t
x0[t]==u[t],​​
∂
t
γ[t]w[t],​​fM[x0[t],γ[t]]*
∂
t
u[t]-fκ[x0[t],γ[t]]*
∂
t
w[t]+
1
2
(
∂
xx
fM[xx,γ[t]]/.xxx0[t])*
2
(
∂
t
x0[t])
-
1
2
(
∂
xx
fm[xx,γ[t]]/.xxx0[t])*
2
(
∂
t
γ[t])
-
∂
gg
fκ[x0[t],gg]/.ggγ[t]*
2
(
∂
t
γ[t])
+
∂
gg
fM[x0[t],gg]/.ggγ[t]*(
∂
t
γ[t])*(
∂
t
x0[t])+(
∂
xx
fV[xx,γ[t]]/.xxx0[t])2π*Γ[t]-α*
2
π
6
*
fF[x0[t]]
γ[t]
*
2
(
∂
xx
fF[xx]/.xxx0[t])
fF[x0[t]]
*(
∂
t
x0[t])-8*α*
γ[t]
fF[x0[t]]
*(
∂
t
x0[t])+α*
2
π
3
*
(
∂
xx
fF[xx]/.xxx0[t])
fF[x0[t]]
*
1
2
γ[t]
*(
∂
t
γ[t]),​​fm[x0[t],γ[t]]*
∂
t
w[t]-fκ[x0[t],γ[t]]*
∂
t
u[t]+
1
2

∂
gg
fm[x0[t],gg]/.ggγ[t]*
2
(
∂
t
γ[t])
-
1
2

∂
gg
fM[x0[t],gg]/.ggγ[t]*
2
(
∂
t
x0[t])
-(
∂
xx
fκ[xx,γ[t]]/.xxx0[t])*
2
(
∂
t
x0[t])
+(
∂
xx
fm[xx,γ[t]]/.xxx0[t])*(
∂
t
γ[t])*(
∂
t
x0[t])+
∂
gg
fV[x0[t],gg]/.ggγ[t]α*
2
π
3
*
(
∂
xx
fF[xx]/.xxx0[t])
fF[x0[t]]
*
1
2
γ[t]
*(
∂
t
x0[t])-α*
2
2
π
3
*
fF[x0[t]]
3
γ[t]
*(
∂
t
γ[t]),​​x0[0]xp,​​u[0]==v0,​​γ[0]
1
1-
2
v0
,​​w[0]0​​,{x0,γ,u},{t,0,tmax},Method"Adams",StepMonitor:>(p=t;++n),{n,p};
In[]:=
<<MaTeX`(*Usedtogenerateaxisdescriptionsongraphs.*)
A comparison of the position of the kink obtained from the PDE model (black line) and the effective model (red dashed line).
ShowDensityPlotSin
π
12
*y,{x,0,tmax},{y,-20,30},ColorFunction(ColorData["Rainbow"][Rescale[#,{-1.3,1.3}]]&),ColorFunctionScalingFalse,PlotPoints500,PlotTheme"Scientific",PlotLegendsPlaced[BarLegend[{Automatic,{-1,1}},LegendMarkerSize190,Ticks{{-1,MaTeX["-1",FontSize14]},{0,MaTeX["0",FontSize14]},{1,MaTeX["+1",FontSize14]}}],Right],PlotRange{{0,400},{-18,30}},FrameStyleDirective[11,FontFamily->"Century Schoolbook",Black],FrameLabel{{MaTeX["\\boldsymbol{x_0(t)}",FontSize18],None},{MaTeX["\\boldsymbol{t}",FontSize18],None}},FrameTicks{{True,True},{True,True}},AspectRatio12,PlotRangePadding0,FrameTicksStyle{{Automatic,Automatic},{Automatic,Automatic}},ListLinePlot[outx0,PlotStyle->{Black,Thickness[0.008]},PlotRange{All,All}],Plot[x0[t]/.model,{t,0,tmax},PlotStyle{Dashed,Thickness[0.007]},PlotTheme"Scientific",PlotRangeAll],Graphics[{Opacity[0.2],Rectangle[{50,-20},{200,30}]}],Epilog{Text[MaTeX["\\boldsymbol{(a)}",FontSize16],Scaled[{.955,.93}]]}
A comparison of the gamma variable obtained from the PDE model (black line) and the effective model (red dashed line).
The trajectory placed on the phase portrait background, also containing the transition from the case with bias (blue line) to the one without (red line).
In this work, we revisited the interaction of a kink with a heterogeneous environment. A key contribution is a modified ansatz that accounts for heterogeneity while mitigating side effects from the unperturbed sine-Gordon kink. We also introduced a variable for the inverse kink width, enabling a significantly more accurate characterization of both the conservative and damped-driven cases. Our approach captures not only localized but also periodic heterogeneities. By tracking the kink’s center and inverse width, we demonstrate a far more precise description than previous studies, so effective that the derived ODE model could potentially replace the PDE over a broad parameter range. Moreover, the periodic landscape offers a means of manipulating the kink’s trajectory and final position by adjusting the external bias drive.

CITE THIS NOTEBOOK

Kink movement on a periodic background​
by Tomasz Dobrowolski, Jacek Gatlik & Panayotis G. Kevrekidis
Wolfram Community, STAFF PICKS, March 19, 2025
​https://community.wolfram.com/groups/-/m/t/3420174